scholarly journals ON HOMOGENEOUS CR MANIFOLDS AND THEIR CR ALGEBRAS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1199-1214
Author(s):  
ANDREA ALTOMANI ◽  
COSTANTINO MEDORI
Keyword(s):  
Lie Group ◽  
Levi Form ◽  
Flag Manifold ◽  
Semisimple Lie Group ◽  
Real Form ◽  
Cr Manifolds ◽  
Geometrical Properties ◽  
Homogeneous Cr Manifolds

In this paper we show some results on homogeneous CR manifolds, proved by introducing their associated CR algebras. In particular, we give different notions of nondegeneracy (generalizing the usual notion for the Levi form) which correspond to geometrical properties for the corresponding manifolds. We also give distinguished equivariant CR fibrations for homogeneous CR manifolds. In the second part of the paper we apply these results to minimal orbits for the action of a real form of a semisimple Lie group Ĝ on a flag manifold Ĝ/Q.

scholarly journals Invariant measures on nilpotent orbits associated with holomorphic discrete series

2021 ◽  
Vol 25 (24) ◽  
pp. 732-747
Author(s):  
Mladen Božičević
Keyword(s):  
Lie Group ◽  
Invariant Measures ◽  
Discrete Series ◽  
Coadjoint Orbits ◽  
Semisimple Lie Group ◽  
Real Form ◽  
Holomorphic Discrete Series ◽  
Content Type ◽  
Wave Front Set ◽  
Complex Semisimple

Let G R G_\mathbb R be a real form of a complex, semisimple Lie group G G . Assume G R G_\mathbb R has holomorphic discrete series. Let W \mathcal W be a nilpotent coadjoint G R G_\mathbb R -orbit contained in the wave front set of a holomorphic discrete series. We prove a limit formula, expressing the canonical measure on W \mathcal W as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the positive chamber defined by the Borel subalgebra associated with holomorphic discrete series.

scholarly journals Blow-ups and infinitesimal automorphisms of CR-manifolds

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1701-1724
Author(s):  
Boris Kruglikov
Keyword(s):  
Lie Group ◽  
Blow Up ◽  
Symmetry Algebra ◽  
Levi Form ◽  
Cr Manifolds ◽  
Parabolic Subalgebras ◽  
Degeneracy Locus ◽  
Real Analytic ◽  
Infinitesimal Automorphisms

Abstract For a real-analytic connected CR-hypersurface M of CR-dimension $$n\geqslant 1$$ n ⩾ 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra $$\mathfrak {s}={\mathfrak {s}}(M)$$ s = s ( M ) : (i) either $$\dim {\mathfrak {s}}=n^2+4n+3$$ dim s = n 2 + 4 n + 3 and M is spherical everywhere; (ii) or $$\dim {\mathfrak {s}}\leqslant n^2+2n+2+\delta _{2,n}$$ dim s ⩽ n 2 + 2 n + 2 + δ 2 , n and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.

A LIMIT FORMULA FOR EVEN NILPOTENT ORBITS

2008 ◽  
Vol 19 (02) ◽  
pp. 223-236
Author(s):  
MLADEN BOŽIČEVIĆ
Keyword(s):  
Lie Group ◽  
Coadjoint Orbits ◽  
Nilpotent Orbits ◽  
Semisimple Lie Group ◽  
Real Form ◽  
Parabolic Subalgebra ◽  
Limit Formula ◽  
Canonical Measure ◽  
Complex Semisimple

Let Gℝ be a real form of a complex, semisimple Lie group G. Assume [Formula: see text] is an even nilpotent coadjoint Gℝ-orbit. We prove a limit formula, expressing the canonical measure on [Formula: see text] as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the negative chamber defined by the parabolic subalgebra associated with [Formula: see text].

scholarly journals A Drift-Free Left Invariant Control System on the Lie Group SO(3)×R3×R3

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Camelia Pop
Keyword(s):  
Control System ◽  
Numerical Integration ◽  
Lie Algebra ◽  
Lie Group ◽  
Poisson Structure ◽  
Dual Space ◽  
Free System ◽  
Geometrical Properties ◽  
Free Left ◽  
Invariant Control

A controllable drift-free system on the Lie group G=SO(3)×R3×R3 is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on g∗,·,·- are studied, where ·,·- is the minus Lie-Poisson structure on the dual space g∗ of the Lie algebra g=so(3)×R3×R3 of G. The numerical integration of this system is also discussed.

scholarly journals Fibrations and globalizations of compact homogeneous CR-manifolds

2009 ◽  
Vol 73 (3) ◽  
pp. 501-553 ◽  
Author(s):  
B Gilligan ◽  
Alan T Huckleberry
Keyword(s):  
Cr Manifolds ◽  
Homogeneous Cr Manifolds
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